Algebraic Geometry Seminar: Jesse Wolfson

Higher stacks arise in many contexts in algebraic geometry and differential topology. The simplest type are higher principal bundles, special cases of which include principal bundles and n-gerbes. Locally, these objects are presentable by higher cocycles on a hypercover. With ordinary principal bundles, we obtain a bundle from a cocycle by using the cocycle to construct a Lie groupoid over the trivial bundle on the cover, and then passing to its orbit space. We establish the existence of an analogous construction for arbitrary higher principal bundles. Unpacking this construction in examples, we recover the familiar definitions of principal bundles, bundle gerbes, multiplicative gerbes and their equivariant versions, now seen as instances of a single construction. Applications beyond this include establishing a representability criterion for connected simplicial presheaves, and a Lie's 3rd theorem for finite dimensional L_oo-algebras.